Michael Weiss: Anschaulichkeit, Abscheulichkeit

My title comes from a famous pair of quotes, about the love lost
between Schroedinger and Heisenberg. The first quote is in a footnote
to Schroedinger's "equivalence" paper, where he proved the equivalence
of his wave mechanics to Heisenberg's matrix mechanics.

My theory was inspired by L. de Broglie and by brief but infinitely
far-seeing remarks of A. Einstein (Berl. Ber. 1925, p. 9ff.) I was
absolutely unaware of any genetic relationship with Heisenberg. I
naturally knew about his theory, but because of the to me very
difficult-appearing methods of transcendental algebra and the lack
of Anschaulichkeit [visualizability], I felt deterred, by it, if not
to say repelled. [1]

Heisenberg responded in a letter to Pauli:

The more I think about the physical portion of the Schroedinger
theory, the more repulsive [abscheulich] I find it....What
Schroedinger writes about the visualizability of his theory 'is
probably not quite right', in other words it's crap. [2]

Most of the philosophical debates swirling around quantum mechanics
have to do with causality. We all know what a wet blanket Einstein
was on Las Vegas night. (Probably still is. "Put those dice DOWN!
I'm talking to you, God!")

But in the childhood of quantum theory, the matter of visualizability
loomed just as large. In his second paper on wave mechanics,
Schroedinger wrote:

...it has even been doubted whether what goes on in an atom can be
described within a scheme of space and time. From a philosophical
standpoint, I should consider a conclusive decision in this sense as
equivalent to a complete surrender. For we cannot really avoid our
thinking in terms of space and time, and what we cannot comprehend
within it, we cannot comprehend at all. There are such things but
I do not believe that atomic structure is one of them. [3]

Schroedinger wrote to Willy Wien:

Bohr's standpoint, that a space-time description is impossible, I
reject a limine... If [atomic research] cannot be fitted into
space and time, then it fails in its whole aim and one does not know
what purpose it really serves. [4]

Bohr and Heisenberg of course held a different opinion. The founding
papers on matrix mechanics expressed the operational philosophy: "You
got your equations, you got your observations, and they match. What
more do you want? Shut up and calculate!" Of course, they had to say
it more politely, at least in print. For example, here's the
abstract, in full, of Heisenberg's famous paper:

The present paper seeks to establish a basis for theoretical quantum
mechanics founded exclusively upon relationships between quantities
which in principle are observable. [5]

And in the introduction to the "Dreimaennerarbeit", the paper that
laid out the whole structure for the first time:

Admittedly, such a system of quantum-theoretical relations
between observable quantities...would labor under the
disadvantage of not being directly amenable to a geometrically
visualizable interpretation, since the motion of electrons
cannot be described in terms of the familiar concepts of
space and time. [6]

but then they immediately point out what really counts: the
equations of motion have the same form as in classical physics. Yet
in the same paragraph, they concede:

In the further development of the theory, an important task will lie
in the closer investigation of the nature of this correspondence and
in the description of the manner in which symbolic quantum geometry
goes over into visualizable classical geometry.

Philosophically, I can't say Schroedinger's position makes sense to
me. Our visual systems sport some pretty complicated circuity, a lot
of it quite non-intuitive. (Read Hubel's Eye, Brain, and Vision for
the nifty details.) It's first rate for watching movies and playing
video games, and similar Cro-Magnon activities. It didn't evolve to
help us understand quantum field theory!

On the other hand, I'm a visualizer myself. I started the "photons,
schmotons" thread with one question in mind: just how far can you
visualize the QFT description of light? I had in mind something like
the balloon analogy in GR: it's wrong because of (a), (b), and (c),
but it's still useful because it does capture (e) and (f). Little did
I know...

Perhaps by the time the infamous, never-ending, hydra-headed "Photons,
Schmotons" thread runs its course, my question will be answered. For
the rest of this post though I want to talk about the discarded
pictures (discarded by QFT, at any rate). Presumably deadly
experimental results could be marshalled to drive stakes through the
hearts of all these alternatives, but I'll leave that for someone else
to discuss. Pretend we've turned up an old family album in the attic.
Each quaint sepia-toned photograph draws our interest and affection.
Someone else can recount how prosperous-looking Uncle Max went
bankrupt in 1926.

OK, let's say we're determined to visualize wave-particle duality,
experiments be hanged! What are our options? I can think of four.

Photons with rhythm

Our photons are little ball-bearings, but with waves etched on the
surface like a designer logo. I am reminded of Newton's light
corpuscles, with their "Fits of easy Reflection" and "Fits of easy
Transmission".

The Ten-Minute History of Science says, "Newton, light
particles---BAD! Huyghens, light waves---GOOD!" It comes as a bit of
surprise to learn that Newton's Opticks is filled with observations
of interference and diffraction phenomena. Newton concluded that his
corpuscles had to undergo a periodic change of state, swinging back
and forth between Fits of easy Reflection and Fits of easy
Transmission.

Newton's theory of light had three characteristics:

particulate nature

periodicity

polarization

(I love the way Newton described polarization: the corpuscles have
Sides.)

How does Huyghens stack up?

wave nature

no periodicity

no polarization

Yes, no periodicity in Huyghens! [7] Surprised me too. Score two out
of three for Newton! [8]

Pilot waves

Of course we think of de Broglie, and this line of thought lead
eventually to Bohm's interpretation of QM. But Newton again takes
priority. The famous non-framer of hypotheses couldn't refrain from
speculating:

...when a Ray of Light falls upon the Surface any pellucid Body, and
is there refracted or reflected, may not Waves of Vibrations, or
Tremors, be thereby excited... and are not these Vibrations
propagated from the point of Incidence to great distances? And do
they not overtake the Rays of Light, and by overtaking them
successively, do they not put them into the Fits of easy Reflexion
and easy Transmission described above? [9]

Continuous waves, discontinuous emission and/or absorption

According to the pop-history of science, Planck's theory fell into
this category. For example:

Imagine a sponge in a bathtub... According to Maxwell, when a sponge
is squeezed it sends out its water in the the usual way and causes
waves in the bathtub. Planck's sponge is of a rarer sort. Indeed
it is more like a bunch of grapes than a sponge, consisting of
myriads of tiny balloons of various sizes, each full of water. When
this sponge is squeezed, the balloons burst one after the other,
each shooting out its contents in a single quick explosion--- a
bundle of water--- and setting up waves... Einstein, however, took
the sponge right out of the bathtub... When he squeezed his sponge
gently, water fell from it like shimmering drops of rain. [10]

However, in 1912 Planck did come up with his so-called "second
theory", in which emission is discontinuous, while propagation and
absorption remain continuous.

Kuhn's book has full details. Though Planck's second theory never
made it to the big time, it did come up with two hits: zero-point
energy made its first appearance here, and Bohr got some inspiration
for his model of the atom.

Wave packets

Early on, Schroedinger suggested that his wavefunction (for an
electron) meant that the charge really was spread out--- space was
pervaded with a kind of electron goop. In his equivalence paper (the
one with the "Anschaulichkeit" footnote), he notes:

There are today not a few physicists who, exactly in the sense of
Mach and Kirchhoff, see the task of physical theory to be merely the
most economial description of empirical connections between
observable quantities... In this view, mathematical equivalence
means almost the same as physical equivalence. [12]

So is matrix mechanics just as good as wave mechanics, or maybe even
better, because it doesn't clutter up the story with fairy tales? No,
say Schroedinger--- physicists need space-time (i.e., pictorial)
descriptions to make progress. He then proposes an interpretation of
the wavefunction psi: the real part of (psi d psi/dt) gives the
spatial density of electric charge.

Schroedinger also constructed a wave-packet: a well-localized psi
function that stays together in time. He did this for a harmonic
oscillator potential (just our coherent states, I'll bet!), but he
hoped originally to do the same in general.

All waves, no particles anywhere! Can it really be that simple?
Schroedinger hoped so.

Schroedinger sent his papers to "grey eminence of theoretical
physics", Hendrik Lorentz. (Lorentz incidentally was the first fellow
to convince Planck that the black-body formula could not be derived
without some sort of discontinuity assumption.)

Lorentz raised several objections [13]. First, he noted that psi is
function of (x,y,z) only in the single-particle case. With two
particles, psi becomes a function of six variables, the coordinates
of both particles:

If I had to choose between wave mechanics and matrix mechanics, I
would give preference to the former because of its greater
Anschaulichkeit, so long as one is concerned only with the
coordinates x,y,z. With a greater number of degrees of freedom,
however, I cannot interpret physically the waves and vibrations in
q-space and I must decide for matrix mechanics.

Lorentz also pointed out that the harmonic oscillator potential was
quite special, and that in the field of a hydrogen atom, the wave
packet would spread out rapidly.

I won't go through the rest of Lorentz's criticisms. Schroedinger's
biographer notes:

Lorentz belonged to an older generation of physicists, and
Schroedinger might have drawn from their discussions the conclusion
that his new discoveries cannot be fitted into a classical framework
at all.

[7] Or so says I. Bernard Cohen in the preface to the Dover edition of
the Opticks (see page xlvii).

[8] Why not three out of three, if we believe in quantum mechanics?
I side with I.Bernard Cohen: "...we must choose between (1) the
historical or (2) the antiquarian approach to the development of
science... the antiquarian's sifting of the disjecta membra of
the Opticks (often out of context) for an occasional
'precursorship' of one or another 20th-century physical concept."
(op. cit.)